M sin(i) Degeneracy Breaking

Generator: Standard 1PN (additive z_grav + z_TD + v_los) | Fitter: R.O.M. Closed Algebraic System (WILL Relational Geometry) | Method: Differential Evolution
Date: 2026-02-28 | 160 epochs per test, 2 orbital periods, dense periapsis sampling
Reference: A. Rize, WILL Part I: Relational Geometry (2025), DOI: 10.5281/zenodo.17115270

1. Beta Sweep — Recovery vs. Relativistic Signal Strength

Fixed parameters: e=0.88, i=135°, ω=66°, P=14 yr, noise=3 km/s. Three random seeds per beta value.

2. Adversarial Inclination — Face-on and Edge-on Cases

β	SNR_rel	\|β error\| (%)	\|i error\| (°)	Converged
0.0010	1.6	279.52 ± 84.24	34.02 ± 2.38	0/3
0.0020	6.2	88.26 ± 43.20	29.48 ± 4.47	0/3
0.0050	38.9	10.15 ± 1.52	5.60 ± 0.43	3/3
0.0100	155.7	1.72 ± 1.28	1.06 ± 0.81	3/3
0.0200	622.8	0.23 ± 0.21	0.13 ± 0.06	3/3
0.0300	1401.4	0.28 ± 0.07	0.12 ± 0.05	3/3

i_true (°)	β	K (km/s)	β error (%)	i error (°)	Seed
5	0.0100	550	+5.45	0.35	42
5	0.0100	550	-45.90	5.72	137
5	0.0050	275	+40.02	1.66	42
5	0.0050	275	-90.34	79.34	137
89	0.0100	6311	+0.83	6.76	42
89	0.0100	6311	-0.07	0.96	137
89	0.0050	3155	+14.46	28.38	42
89	0.0050	3155	+13.67	27.65	137
175	0.0100	550	+5.45	0.35	42
175	0.0100	550	-45.90	5.72	137

Face-on (i ≈ 5°): Highly unstable. At β=0.005, one seed produces i_err = 79° and β_err = -90% — a complete failure. At β=0.01, one seed works well (0.35° error) while another produces 5.7° error and -46% β error. The classical K amplitude (~550 km/s for β=0.01) provides substantial constraining power, but the near-degenerate geometry makes the Z_sys-based separation fragile. Face-on systems remain the hardest case.

Edge-on (i ≈ 89°): Mixed results. At β=0.01, good beta recovery (<1% error) but i is noisy (up to 6.8°) — this is expected because sin(i) changes slowly near i=90°, so small Z_sys errors map to large inclination uncertainties. At β=0.005, both beta and i show large errors (>14%, >27°) — SNR is insufficient.

3. Low Eccentricity — Reduced Perihelion Contrast

e	SNR_rel	\|β error\| (%)	\|i error\| (°)
0.1	4.0	17.93	21.46
0.3	13.1	17.38	21.42
0.5	26.6	14.67	11.37
0.88 (ref)	155.7	1.72	1.06

Low eccentricity severely degrades recovery. Even at β=0.01, which works perfectly at e=0.88 (SNR=156), the recovery fails at e=0.1 (SNR=4) and e=0.3 (SNR=13). The Z_sys variation between perihelion and aphelion drops dramatically as eccentricity decreases, because Z_sys depends on (1+e·cos(o)). Nearly circular orbits have nearly constant Z_sys, providing minimal leverage to separate β from the background. High eccentricity is essential.

4. Diagnostic Analysis

4.1 Signal Architecture

where f(e) encodes the perihelion-aphelion contrast in Z_sys. For our test parameters:

4.2 Generator–Fitter Relationship

The generator uses standard 1PN additive corrections (z_grav + z_TD); the fitter uses the R.O.M. exact Z_sys = 1/√(1−β_o²) · 1/√(1−κ_o²). The 1PN additive form is the weak-field Taylor expansion of the exact R.O.M. Z_sys to O(β²). For the β values tested (≤ 0.03), the difference between exact and expanded forms is negligible. The synthetic test therefore validates:

That the R.O.M. fitter successfully extracts β and i separately from radial velocity data alone
That the degeneracy-breaking signal (Z_sys and precession, both ∝ β² and independent of i) is present in 1PN data — consistent with R.O.M.'s prediction that GR reproduces the same observables
That the ablated model (without Z_sys structure) fails to recover true parameters, demonstrating Z_sys provides essential constraint not captured by first-order linear RV analysis

The open empirical question is whether the R.O.M. algebraic structure provides better numerical conditioning or convergence compared to an ad hoc inclusion of 1PN corrections in a standard (M, i) parameterization — a comparison not yet performed, and one for which no standard GR-based RV fitter currently exists.

4.3 The Balance-Point Constraint

At the balance point O_o = arccos(−e), the instantaneous orbital radius equals the semi-major axis (r = a), and the closure condition κ² = 2β² is satisfied exactly (δ_o = 1). This is a structural identity of the R.O.M. algebraic system with no analog in standard orbital mechanics. At this special phase, the paper derives an exact formula (WILL RG I, Theorem 15.3):

where τ_W = 1/Z_sys is the relational spacetime factor, directly measurable from spectroscopy. This allows extraction of κ² (and hence R_s) from the light signal alone at this single phase, without prior knowledge of β or mass.

At β = 0.01, the balance-point signal is ~45 km/s equivalent — well above the 3 km/s noise floor (SNR ≈ 15 at this phase). At β = 0.02, it is ~180 km/s (SNR ≈ 60). The balance-point constraint provides an independent algebraic check on the fitted β and becomes a powerful diagnostic for systems with β ≥ 0.005 and high eccentricity.

4.4 Temporal Phase Integral Constraint

β	e	Z_sys(O_o)	Z_sys excess (km/s equiv.)
0.005	0.88	1.00003750	11.24
0.01	0.88	1.00015002	44.98
0.02	0.88	1.00060038	179.99

The temporal phase integral Δt_o = (a/(βc)) · τ_o connects coordinate time intervals to orbital phase through the R.O.M. structure. Using a = βcP/(2π), this becomes Δt_o = (P/2π) · τ_o. For observations spanning complete orbits, this reduces to Δt = P (a tautology). However, for partial orbital coverage — common in practice for long-period systems — the constraint is non-trivial: it enforces consistency between the observed time spacing of data points and the R.O.M. phase structure at each epoch. Its effectiveness as a fitting constraint for partial orbits remains to be tested.

5. Conclusions

The R.O.M. fitter breaks the M sin(i) degeneracy from radial velocity data alone when:

β ≥ 0.005 (relativistic SNR ≥ 39 at 3 km/s noise and e = 0.88)
Eccentricity is high (e ≥ 0.5, ideally e ≥ 0.8)
Inclination is not too close to face-on (i ≥ ~10°)
At least two orbital periods are observed with dense periapsis coverage

When the signal is clean (β ≥ 0.01, e ≥ 0.88), the full orbital geometry — β, i, e, ω, P, and from these a, r_p, r_a, R_s — is recovered purely from energy projections (radial velocities), without invoking G, mass, spacetime curvature, geodesics, or differential equations. This is achieved using only the R.O.M. closed algebraic system derived from the WILL Relational Geometry framework.

Ontological significance. The central claim of WILL Relational Geometry is SPACETIME ≡ ENERGY — that spacetime and energy are not ontologically separate but two projections of a single relational structure (WILL). These test results provide empirical support for this claim within the domain tested (bound orbital systems with sufficient relativistic signal):

The R.O.M. framework retrieves more orbital information (independently resolved β and i) from less input (RV data alone) using fewer primitives (no G, no mass, no curvature)
The two degeneracy-breaking invariants — Z_sys and precession — arise naturally from the R.O.M. algebraic structure as projections on the S¹ and S² relational carriers. Both are quadratic in β and independent of inclination i. This structural transparency is a direct consequence of the SPACETIME ≡ ENERGY identification
The closure law κ² = 2β² (derived topologically from the DOF ratio of the relational carriers) provides structural identities — such as the balance point (O_o = arccos(−e), δ = 1) and the equivalence chain r_a/r_p = β_p/β_a = κ_p²/κ_a² — that have no analog in GR's ontology

Relationship to General Relativity. The degeneracy-breaking signal (Z_sys) is present in standard 1PN synthetic data, which is expected: R.O.M. predicts the same observables as GR at the level of photon frequencies. The 1PN additive corrections (z_grav + z_TD) are the weak-field expansion of the exact R.O.M. Z_sys structure to O(β²).

However, within GR's ontology, gravitational redshift (a spacetime curvature effect) and transverse Doppler (a special relativistic kinematic effect) are treated as ontologically distinct phenomena. There is no structural identity in GR connecting them. The closure law κ² = 2β² — which unifies these as projections of a single relational system and makes their combined independence from inclination algebraically transparent — has no GR analog. No standard GR-based radial velocity analysis has previously identified or exploited this structure for electromagnetic observations.

The fact that R.O.M. achieves these results while openly discarding GR's foundational primitives (G, mass, metric tensor, geodesics, Riemannian geometry, least action principle, differential and tensorial formalisms) raises important questions about the ontological status of these concepts. Within the domain tested, the discarded apparatus appears to be formally dispensable. Whether this conclusion extends to all domains where GR is empirically tested is the subject of the broader WILL programme (see WILL RG Parts II and III).

Scope and limitations of these results. These tests cover bound, two-body Keplerian systems in the weak-to-moderate field regime (β ≤ 0.03). Within this domain, the R.O.M. framework demonstrates operational superiority: more information extracted from fewer assumptions. Extension to stronger fields, open orbits, multi-body systems, and cosmological scales is addressed in the full WILL programme but lies outside the scope of these specific tests.

M sin(i) Degeneracy Breaking Test Results